Finite field/Smooth projective curve/Vector bundle/Strongly semistable degree nonnegative/Finite annihilation/Fact

Let ${}K$ denote a finite field (or the algebraic closure of a finite field) and let ${}X$ be a smooth projective curve over ${}K$. Let ${}{\mathcal {S}}$ be a strongly semistable locally free sheaf over ${}X$ of nonnegative degree and let ${}c\in H^{1}(X,{\mathcal {S}})$ denote a cohomology class.

Then there exists a finite morphism

$\varphi \colon Y\longrightarrow X$

such that ${}\varphi ^{*}(c)$ is trivial.

Proof, Write another proof